Received: 19 June 2018 Accepted: 07 August 2018 DOI: 10.1002/pamm.201800268 Effect of grain orientations on the thermal grain boundary grooving in a three-dimensional setting Asim Ullah Khan 1, * , Klaus Hackl 1, , and Mattias Baitsch 1, 1 Ruhr-Universität Bochum, Institute of Mechanics of Materials The objective of this work is to study the effect of grain orientation on the thermal grooving by surface diffusion. Hackl et al. [1] have presented a finite element model for thermal grooving in three-dimensions. This variational model involves surface energy, grain boundary energy, external and internal triple line energy. In this study, We use an orientation dependent surface energy. For {1 0 0} grain orientation in the normal direction, we have self-similar groove profiles for increasing extent of anisotropy of the surface energy. For {1 1 0} and {1 1 1} orientations, there are formation of facets for critical anisotropic cases. These formations are due to so-called missing orientations concerning the shape of an unconstrained crystal in equilibrium. The rate of grooving varies with change in the extent of surface free energy anisotropy. Flux along the triple line is also important in determining the groove root shape. Triple line energy and its mobility lead to deviate from a typical t 1 4 scaling law. For all theses simulations, grain boundary energies are constant, satisfying Herring’s relation. Comparisons are made for different values of mobilities for groove shape and its growth rate, using different grain orientations. c  2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Variational model for grooving Variational model for thermal grooving using the principle of maximum dissipation [3] has already been presented in [1]. The following equations are presented here for convince. The Lagrange functional reads L = Q + λ 1  Q + ˙ G  + β j . (other constraints) . (1) In this model, the dissipation Q is considered due to two processes include surface diffusion and diffusion along the groove root. Second term, rate of Gibbs energy, ˙ G has contribution from the thermodynamics forces, which include surface energy γ (n), which is functional of surface normal vector (n), grain boundary energy γ η , triple line energy γ Γ , and internal triple line energy γ Σ multiplied with thermodynamics fluxes, which are functional of relative velocities. The other constraint in the Lagrange includes the conservation of mass in the system. Variation of equation 1 with respect to kinetic variables, gives following set of evolution equations 1 On surface ω i ∈ S : j i = -M i ∇ s μ i , v n,i + ∇ s · j i =0. (2) For each triple line Γ i ∈ T : j Γj = -M Γi dμ Γi ds i , γ ηi + γ Γi K Γi =  j,ωj ∈AΓ i γ j cos α ij , dj Γi ds i +  j j j · m Γij =0. (3) As our model have contribution from triple line energy, thus we get a relation that relates triple line, grain boundary, and surface energies along the meeting boundaries. At the quadruple point Q i ∈ Q :  j,j∈Api s Σij j Γj =0, γ Σi =  j,Γi∈Api γ Γj cosα Γij . (4) Equation 4 shows that there is no net flow of flux at meeting point of all boundaries and also this point is in thermodynamic equilibrium. These sets of coupled equations are solved using finite element method, derived in the Appendix A of [1]. For surface energy anisotropy, simple anisotropic surface energy form is used [4]. γ (n)= γ 0 ( 1 - a ( n 4 1 + n 4 2 + n 4 3 )) where a [0, 1] (5) Figure 2 shows the vector plot for surface energy for different values of scalar factor  a  . The system has minimum value of energy (1 - a) along the principle axis of the faces. With an increase in the value of a, we have formation of ears along the diagonals, which cause the missing orientations in the crystal reference system. When the orientation of crystal sample system lies in the range of these missing orientations, we have formation of flat surfaces along the grain boundaries. These formations are due to high variation of chemical potential along the surface. In this work, we have taken a small rectangular patch shown in figure 1, representing periodic structure of the three regular hexagonal grains. * Corresponding author: e-mail asim.khan@rub.de, phone +49 234 322 6025, 1 For the definition of j surface flux, j flux along the triple line, μ i chemical potential see [1] PAMM · Proc. Appl. Math. Mech. 2018;18:e201800268. www.gamm-proceedings.com c  2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 of 2 https://doi.org/10.1002/pamm.201800268